The effect of temperature on the dynamics of the Repressilator was studied by analysing the functionality, robustness and period of oscillations from the time-series images of oscillating cells, in different temperatures. The statistical methods used in the study are as follows.
5.4.1 Functionality and estimation of periods
The functionality of the masked cells containing the Repressilator was determined using the criterion proposed in (Elowitz et al., 2000). To do so, a fast Fourier transform was applied to the temporal fluorescence signal from each cell and divided by the transform of a decaying exponential with a time constant of 90 min, which is the measured lifetime of the green fluorescent protein used (GFPaav) as the reporter. Cells were classified as oscillatory when the power spectra produce peaks 4.5 times higher than the background at frequencies of 0.15–0.5 per hour. A larger bandwidth, compared to the bandwidth used in (Elowitz et al., 2000), was used here, so as to include failed oscillations, which seemed to increase the period of apparent oscillations by nearly double. The apparent period, for cells that were considered functional, was estimated as follows. First, a quadratic curve in the least-squares sense was fitted to the intensity of the time series to estimate the general trend, since the measured intensity is known to be affected by different factors such as, e.g., photo bleaching. The estimated trend was subtracted from the time series and the residual was scaled to unit power, followed by computation of the autocorrelation function. From this function, the period of oscillation was estimated by locating the first and the third zeros of the autocorrelation function, and computing the distance between them (Fig. 12).
corresponding mean intensities extracted from these cells (bottom). The vertical dashed lines indicate the time points corresponding to the images.
5.4 Statistical analysis
The effect of temperature on the dynamics of the Repressilator was studied by analysing the functionality, robustness and period of oscillations from the time-series images of oscillating cells, in different temperatures. The statistical methods used in the study are as follows.
5.4.1 Functionality and estimation of periods
The functionality of the masked cells containing the Repressilator was determined using the criterion proposed in (Elowitz et al., 2000). To do so, a fast Fourier transform was applied to the temporal fluorescence signal from each cell and divided by the transform of a decaying exponential with a time constant of 90 min, which is the measured lifetime of the green fluorescent protein used (GFPaav) as the reporter. Cells were classified as oscillatory when the power spectra produce peaks 4.5 times higher than the background at frequencies of 0.15–0.5 per hour. A larger bandwidth, compared to the bandwidth used in (Elowitz et al., 2000), was used here, so as to include failed oscillations, which seemed to increase the period of apparent oscillations by nearly double. The apparent period, for cells that were considered functional, was estimated as follows. First, a quadratic curve in the least-squares sense was fitted to the intensity of the time series to estimate the general trend, since the measured intensity is known to be affected by different factors such as, e.g., photo bleaching. The estimated trend was subtracted from the time series and the residual was scaled to unit power, followed by computation of the autocorrelation function. From this function, the period of oscillation was estimated by locating the first and the third zeros of the autocorrelation function, and computing the distance between them (Fig. 12).
Figure 12: Estimation of the period of the Repressilator from the fluorescence intensity signal.
The top panel shows the raw signal extracted from the confocal images, the dashed line show the estimated trend. In the middle panel, the trend from the raw signal was subtracted and the residual was scaled to unit power. The bottom panel shows the autocorrelation function of the treated signal. The distance between the first and the third zeros corresponds to the period of oscillation.
5.4.2 Estimation of robustness and true period of oscillations
In time series, at temperatures over 30 °C, it was observed that a certain fraction of cells, in each condition, produced oscillations followed by a brief period of no activity, in terms of fluorescence levels, which was then followed by resumption of oscillations. This can be due to, either, the failure of the reporter to produce GFP to report an oscillation or expression of reporter is repressed. However, in most cases, the signal from the reporter recovered, following the period of inactivity, thus suggesting failure of oscillations followed by recovery. The method, for estimation of the period of the Repressilator, mentioned above, relies on robust periodic behaviour and it cannot detect if a Repressilator halts its activity for a certain time and then resumes its activity. Instead, it
Figure 12: Estimation of the period of the Repressilator from the fluorescence intensity signal.
The top panel shows the raw signal extracted from the confocal images, the dashed line show the estimated trend. In the middle panel, the trend from the raw signal was subtracted and the residual was scaled to unit power. The bottom panel shows the autocorrelation function of the treated signal. The distance between the first and the third zeros corresponds to the period of oscillation.
5.4.2 Estimation of robustness and true period of oscillations
In time series, at temperatures over 30 °C, it was observed that a certain fraction of cells, in each condition, produced oscillations followed by a brief period of no activity, in terms of fluorescence levels, which was then followed by resumption of oscillations. This can be due to, either, the failure of the reporter to produce GFP to report an oscillation or expression of reporter is repressed. However, in most cases, the signal from the reporter recovered, following the period of inactivity, thus suggesting failure of oscillations followed by recovery. The method, for estimation of the period of the Repressilator, mentioned above, relies on robust periodic behaviour and it cannot detect if a Repressilator halts its activity for a certain time and then resumes its activity. Instead, it
in which cases the measured time became double compared to other oscillations. To extract the true period, a method based on the fact that distribution of period lengths resemble a bimodal distribution when failure of oscillations occur was used (Fig. 13). Here, the mean and standard deviation of the true period in the population was estimated followed by estimation of the fraction of failed oscillations from the measured periods from each cell. For that, the maximum likelihood estimates for a single Gaussian and a mixture of two Gaussians was determined (from mean and standard deviation of the measured periods), such that the mean and the variance of the second are double than that of the first, found using an iterative expectation maximization algorithm (Dempster et al., 1977). Appropriate models were selected using a likelihood ratio test with significance level of 0.01 between the two models, that is the model is selected only when the p-value of a 2-gaussian model is smaller than 0.01. Finally, the fitting was performed, using a leave-one-out-technique, with each subset of data lacking one of the measured periods, which results in N estimates, each using N-1 measure periods from which the variance of the estimates was calculated.
Figure 13: Distribution of periods (magnitude scaled to represent probability density) for each
temperature. Solid lines represent the probability densities of the fitted model with one or two Gaussians. Dashed lines represent the densities of individual components in the case of two Gaussians. For 28 °C and 30 °C, the p-values of the likelihood ratio tests are 0.08 and 1, respectively, indicating a lack of evidence for the two-Gaussian model, whereas for 33 °C and 37
in which cases the measured time became double compared to other oscillations. To extract the true period, a method based on the fact that distribution of period lengths resemble a bimodal distribution when failure of oscillations occur was used (Fig. 13). Here, the mean and standard deviation of the true period in the population was estimated followed by estimation of the fraction of failed oscillations from the measured periods from each cell. For that, the maximum likelihood estimates for a single Gaussian and a mixture of two Gaussians was determined (from mean and standard deviation of the measured periods), such that the mean and the variance of the second are double than that of the first, found using an iterative expectation maximization algorithm (Dempster et al., 1977). Appropriate models were selected using a likelihood ratio test with significance level of 0.01 between the two models, that is the model is selected only when the p-value of a 2-gaussian model is smaller than 0.01. Finally, the fitting was performed, using a leave-one-out-technique, with each subset of data lacking one of the measured periods, which results in N estimates, each using N-1 measure periods from which the variance of the estimates was calculated.
Figure 13: Distribution of periods (magnitude scaled to represent probability density) for each
temperature. Solid lines represent the probability densities of the fitted model with one or two Gaussians. Dashed lines represent the densities of individual components in the case of two Gaussians. For 28 °C and 30 °C, the p-values of the likelihood ratio tests are 0.08 and 1, respectively, indicating a lack of evidence for the two-Gaussian model, whereas for 33 °C and 37
°C, the p-values are 0.0065 and 0.0015, respectively, indicating that the two-Gaussian model should be favoured over the one-Gaussian model.
5.4.3 Loss of synchrony
The effect of temperature on the loss of synchrony between sister cells was studied by detecting pairs of sister cells in which both of the sister cells were functional, and then manually following them by the criterion mentioned above. The loss of synchrony was estimated by calculating the correlation coefficients between functional sister cells. First, the likeliness of a robust cell to produce a robust sister cell, at 37 °C, was obtained. About 112, 40, and 35 pairs were found where none, one, or both of the cells were robust, respectively, suggesting that the numbers of pairs where either none or both sisters are robust are overrepresented. More specifically, there is about a 0.64 chance for a robust cell to have a robust sister, and about a 0.85 chance for a non-robust cell to have a non- robust sister (cf. 0.37). The significance of the correlations was confirmed by calculating the p-value of one-tailed Fisher’s exact test with the null hypothesis. It suggests that being robust or not is independent of the sister cells, resulting in a p-value smaller than 1.29×10-10.Next, we computed the correlation between the intensity signals of each pair of sister cells. This correlation results from loss of synchrony caused both by division and variations in the behaviour of the cells over their lifetime (i.e. variations/drift in the period and noise in the intensity signal). The distributions of correlation coefficient extracted from each pair of cells are shown in Fig. 14, and the mean and standard deviation of the coefficients are shown in Table 4.
It was found that, on average, as temperature increases, sister cells lose correlation (p- values of 3.17×10-3 and 9.90×10-4 for 28 vs. 30 °C and 30 vs. 37 °C in one-tailed Welch’s t-test with the null hypothesis that the means are equal). This loss of correlation may be due to the increase in the noise of the period as a function of the temperature, which appears to follow a similar pattern. Accordingly, since the non-robust cells contribute much of the variation in the 37 °C condition, the correlation is restored to a level comparable to the 30 °C condition when only the robust cells are considered. Interestingly, Fig. 14 reveals that, in each condition, most cells are very highly correlated.
°C, the p-values are 0.0065 and 0.0015, respectively, indicating that the two-Gaussian model should be favoured over the one-Gaussian model.
5.4.3 Loss of synchrony
The effect of temperature on the loss of synchrony between sister cells was studied by detecting pairs of sister cells in which both of the sister cells were functional, and then manually following them by the criterion mentioned above. The loss of synchrony was estimated by calculating the correlation coefficients between functional sister cells. First, the likeliness of a robust cell to produce a robust sister cell, at 37 °C, was obtained. About 112, 40, and 35 pairs were found where none, one, or both of the cells were robust, respectively, suggesting that the numbers of pairs where either none or both sisters are robust are overrepresented. More specifically, there is about a 0.64 chance for a robust cell to have a robust sister, and about a 0.85 chance for a non-robust cell to have a non- robust sister (cf. 0.37). The significance of the correlations was confirmed by calculating the p-value of one-tailed Fisher’s exact test with the null hypothesis. It suggests that being robust or not is independent of the sister cells, resulting in a p-value smaller than 1.29×10-10.Next, we computed the correlation between the intensity signals of each pair of sister cells. This correlation results from loss of synchrony caused both by division and variations in the behaviour of the cells over their lifetime (i.e. variations/drift in the period and noise in the intensity signal). The distributions of correlation coefficient extracted from each pair of cells are shown in Fig. 14, and the mean and standard deviation of the coefficients are shown in Table 4.
It was found that, on average, as temperature increases, sister cells lose correlation (p- values of 3.17×10-3 and 9.90×10-4 for 28 vs. 30 °C and 30 vs. 37 °C in one-tailed Welch’s t-test with the null hypothesis that the means are equal). This loss of correlation may be due to the increase in the noise of the period as a function of the temperature, which appears to follow a similar pattern. Accordingly, since the non-robust cells contribute much of the variation in the 37 °C condition, the correlation is restored to a level comparable to the 30 °C condition when only the robust cells are considered. Interestingly, Fig. 14 reveals that, in each condition, most cells are very highly correlated.
However, increases in the temperature results in pairs of cells with wider range of correlation coefficients, including a sizable number of pairs whose series are strongly anticorrelated.
Figure 14: Distributions of correlation coefficients between functional sister cells in 28 (top), 30
(middle), and 37 °C (bottom). In the 37 °C condition, the pairs where both cells are robust are represented in dark gray bars, while the others are represented in light gray.
Statistic 28 °C 30 °C 37 °C 37 °C,
robust
Correlation
mean 0.87 0.66 0.42 0.63
Correlation s.d 0.16 0.58 0.66 0.58
Table 4: Correlation between sister cells in various temperatures.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 n u m b e r o f pa ir s -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 50 100 correlation coefficient
However, increases in the temperature results in pairs of cells with wider range of correlation coefficients, including a sizable number of pairs whose series are strongly anticorrelated.
Figure 14: Distributions of correlation coefficients between functional sister cells in 28 (top), 30
(middle), and 37 °C (bottom). In the 37 °C condition, the pairs where both cells are robust are represented in dark gray bars, while the others are represented in light gray.
Statistic 28 °C 30 °C 37 °C 37 °C,
robust
Correlation
mean 0.87 0.66 0.42 0.63
Correlation s.d 0.16 0.58 0.66 0.58
Table 4: Correlation between sister cells in various temperatures.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 n u m b e r o f pa ir s -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 50 100 correlation coefficient